#6437184

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1401 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,31,33)(27,29,34,30)(28,36,38,35)(32,39,40,37);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,30,26,31,34,29,33)(28,36,37,40,38,35,39,32);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,30,37,31,38,29,39)(26,32,34,36,33,40,27,35);(10,15)(14,20)(16,22)(21,24)(26,33)(27,34)(32,40)(35,36);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(25,29,31,30)(26,34,33,27)(28,37,38,39)(32,35,40,36)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308959, #72308988
maximal normal subgroup of	#72308959, #72308988

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6344380);